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4 Developmental Variation, Sociocultural Influences, and Difficulties in Mathematics There is evidence that most children bring foundational resources and knowledge about mathematics to school. However, this is not the whole story. Research findings reveal enormous discrepancies in young childrenâs levels of mathematics competence, and these discrepancies appear to be larger in the United States than they are in some other countries (e.g., China) (Starkey and Klein, 2008). This chapter describes the kinds of dif- ferences that exist and reviews what is known about the nature and sources of developmental variations among children. Most children bring core number sense or number competencies to school (National Research Council, 2001). Number sense refers to in- terconnected knowledge of numbers and operations. Although preverbal number sense begins in infancy and appears to be universal, preschool and kindergarten number sense involves understanding of number words and symbols, which is heavily influenced by experience and instruction. The number sense children bring to kindergarten is highly predictive of their later mathematics achievement. The term ânumber senseâ means different things in different fields of research, and almost no two researchers define it in exactly the same way (Gersten, Jordan, and Flojo, 2005; Jordan et al., 2006). The term ânumber senseâ is used in this chapter because much of the research summarized here uses it. When the discussion is more general, the term ânumber competenciesâ is used along with number sense to remind the reader that we are talking about knowledge and skills that can be taught and learned. The word âcompetenciesâ is used as a balanced term mean- ing both knowledge and skills. The competencies encompassed by the term ânumber senseâ as used here are described more fully in Chapter 5. 95
96 MATHEMATICS LEARNING IN EARLY CHILDHOOD Despite strong universal starting points, striking individual differences in number sense emerge early in life and are present by the time children enter preschool (e.g., Klibanoff et al., 2006). These differences are apparent both on standardized tests (e.g., Arnold et al., 2002; Starkey, Klein, and Wakeley, 2004) and on specific measures tapping early number competen- cies, such as determining set size, comparing sets, and carrying out calcu- lations (e.g., Entwisle and Alexander, 1990; Ginsburg and Russell, 1981; Griffin, Case, and Siegler, 1994; Jordan, Huttenlocher, and Levine, 1992; Levine et al., in preparation; Saxe, Guberman and Gearheart, 1987). The level of number sense in kindergarten is highly of predictive future math- ematics success in first through third grades (Fuchs et al., 2007; Jordan, Glutting, and Ramineni, in press; Locuniak and Jordan, in press; Mazzocco and Thompson, 2005) as well as into the later school years (Duncan et al., 2007). In this chapter, we explore individual differences in childrenâs math- ematics competence. We begin by describing the differences associated with key social groups specifically defined by socioeconomic status, gender, race/ethnicity, and English language ability. We then discuss the contextual factors and early experiences that appear to be linked to these differences, giving particular attention to the role of the family and language. We then discuss learning disabilities. We end with a brief discussion of potential intervention. GROUP DIFFERENCES IN MATHEMATICS PERFORMANCE Researchers have explored several key social factors that are linked to systematic, average differences in childrenâs mathematical performance. Socioeconomic status (SES), which includes income level as well as level of parental education, is strongly linked to differences in mathematics com- petence. Evidence related to gender differences in mathematics competence is less clear, although some differences have been found. Socioeconomic Status Mathematical skills of young children from low-income families lag behind those of their middle-income peers. Preschoolers who attend Head Start Programs perform significantly below children who attend preschools serving middle-income children on standardized tests of mathematical read- iness (Ehrlich and Levine, 2007). The gulf between low- and middle-income children is wide and includes spatial/geometric and measurement as well as number competencies (Clements, Sarama, and Gerber, 2005; Klein and Starkey, 2004; Saxe et al., 1987). Jordan and colleagues (Jordan et al., 2006, 2007) found that low-
DEVELOPMENTAL VARIATION, SOCIOCULTURAL INFLUENCES 97 income children enter kindergarten far behind their middle-income peers on tasks assessing counting skills, knowledge of number relations (e.g., recognizing which number is smaller), and number operations. Moreover, longitudinal assessment over six data points revealed that low-income chil- dren were four times more likely than their middle-income peers to show flat growth in these areas throughout kindergarten and early first grade. Underlining the importance of early number sense to school success, the researchers found that level of performance on a battery assessing number sense in kindergarten as well as rate of growth between kindergarten and first grade accounted for 66 percent of the variance in mathematics learn- ing at the end of first grade (Jordan et al., 2007). In other words, number sense in kindergarten in strongly related to competence in mathematics at the end of first grade and the rate of growth over the first grade year. In- come status, gender, age, and reading ability did not account for additional variance in first grade mathematics outcomes over and above initial per- formance and growth in number sense. This suggests that SES differences found at the end of first grade are due to initial differences in number sense in kindergarten. Several studies indicate that SES differences in preschoolersâ number skills are more marked on tasks tapping number skills without objects (called ver- bal tasks) than on tasks tapping number skills with objects (called nonverbal tasks). When kindergarten and first grade children are presented with verbal calculation problems with no objects, either as number combination prob- lems (âHow much is 3 and 2?â) or story problems (âMike had 3 pennies. Jen gave him 2 more pennies. How many pennies does Mike have now?â), middle-income children perform much better than do low-income children (Jordan et al., 2006; Jordan, Huttenlocher, and Levine, 1992; Jordan, Levine, and Huttenlocher, 1994). Middle-income children also achieve at a faster rate on calculation problems without objects in kindergarten (Jordan et al., 2006, 2007). In contrast, SES differences are smaller if the same calculations are presented in a nonverbal format with objects (e.g., the child is shown 3 disks that are then hidden with a cover. The tester then slides 2 disks under the cover and the child indicates how many are now hidden). Jeong and Levine (2005) have shown that knowing number words is associated with very early performance on numerosity matching tasks that do not require verbal responses (e.g., matching arrays of visual dots). Spe- cifically, performance on these tasks is more exact for children who have acquired the meaning of a few number words. For instance, 2- to 3-year- olds were more exact in their ability to match small set sizes when they have better knowledge of the cardinal meanings of number words. Although low-income children performed worse than middle-income children on such numerosity matching tasks, this difference was eliminated if answers that were plus or minus 1 from the correct answer were counted as correct
98 MATHEMATICS LEARNING IN EARLY CHILDHOOD (Ehrlich, Levine, and Goldin-Meadow, 2006). Thus, low-SES preschoolers appear to have approximate representations of set sizes and number words at a time when their higher SES peers have gained exact representations. Therefore, low-SES preschoolers need experiences to learn number words and to use them to help on these matching tasks. The sources of these differences are difficult to pinpoint. Research on childrenâs early experiences point to the amount of support for mathematics at home as well as other language and contextual factors. Some findings show that young children from low-income families receive less support for mathematics in their home environment than do their middle-income peers (Blevins-Knabe and Musun-Miller, 1996; Holloway et al., 1995; Saxe et al., 1987; Starkey et al., 1999). Compounding the situation, public preschool programs serving low-income families tend to provide fewer learning op- portunities and supports for mathematical development than ones serving middle-income families (Clements and Sarama, 2008). These factors are discussed in greater detail in the section on the influence of context and experience. Gender Results and opinions vary regarding gender differences in early math- ematics. Some studies have no revealed gender differences in mathematics performance (e.g., Clements and Sarama, 2008; Lachance and Mazzocco, 2006; Levine, Jordan, and Huttenlocher, 1992; Sarama et al., 2008). Some have found differences favoring boys: Jordan et al. (2006) found small but statistically significant gender effects on calculation with objects and on numerical estimation. In particular, boys had an edge over girls even when income level, age, and reading ability were controlled for in the analyses, and there were more boys than girls in the highest performing group. How- ever, Coleyâs (2002) analysis of the Early Childhood Longitudinal Study da- tabase indicated small advantages in kindergarten in different areas for each gender: Girls were somewhat better in recognizing numbers and shapes, and boys were somewhat better in numerical operations. Some research with older children indicates that girls in the primary grades may tend to use less advanced strategies than do boys (Fennema et al., 1998), and other work suggests no gender differences in the math- ematics performance of older students (Hyde et al., 2008). Recent research (e.g., Carr et al., 2007) suggests spatial skills may promote the use of more advanced computational strategies, and boys seem to have an advantage in the more general area of spatial cognition, even in preschool. There are differences in the mean level of performance of boys and of girls on mental rotation tasks by 4Â½ years of age, ranging from small but significant dif- ferences (Levine et al., 1999) to large differences with girls performing at
DEVELOPMENTAL VARIATION, SOCIOCULTURAL INFLUENCES 99 chance levels (Rosser et al., 1984). Preschool boys also perform better than preschool girls on solving problems involving mazes (e.g., Fairweather and Butterworth, 1977; Wechsler, 1967; Wilson, 1975) and are faster at copy- ing a three-dimensional Lego (plastic blocks) model (Guiness and Morley, 1991). However, it appears that at least some of these differences are cre- ated by lack of particular types of experiences (Ebbeck, 1984). Spatial skill may reflect or at least interact with greater engagement of boys than girls in spatial activities, such as building with Legos (Baenninger and Newcombe, 1989). Young boys typically spend more time playing with Legos and putting puzzles together than do girls, suggesting that engage- ment in spatial activities promotes skill development (Levine et al., 2005). The amount of puzzle play for both boys and girls was related to the men- tal transformation performance (McGuinness and Morley, 1991). Parentsâ spatial language may be more important for girls than for boys; use of such language by parents related to mental transformation performance of girls but not of boys (Cannon, Levine, and Huttenlocher, 2007). Boys tend to be more interested in movement and action from the first year of life and girls more focused on social interactions (e.g., Lutchmaya and Baron-Cohen, 2002). Boys also may gesture more on spatial tasks (e.g., Ehrlich, Levine, and Goldin-Meadow, 2006), indicating that encouraging gesture, especially for girls, may be helpful in spatial learning. Given the finding that boys seem to have an advantage in spatial cog- nition and that this seems to result partly from the number of experiences they have that support such learning, it seems particularly important for both numerical and spatial learning that girls be given opportunities for spatial learning. Importantly, intervention studies with preschoolers using a research-based mathematics curriculum did not find an interaction with gender, indicating that girls can learn as much as boys in both numeri- cal and spatial tasks (Clements and Sarama, 2008; Sarama et al., 2008). Simple modifications to everyday preschool activities, such as block build- ing (Kersh, Casey, and Young, in press) and the use of stories about spatial topics (Casey et al., 2008), have been shown to be effective in developing girlsâ spatial cognition. Teachers should ensure that girls play with blocks and provide them with challenges that ensure that they extend their block- building skills, such as building windows, bridges, and arches. Race and Ethnicity Over the past several decades, research has found differences in chil- drenâs mathematics learning outcomes as a function of their race/ethnicity (e.g., Ginsburg and Russell, 1981). This section discusses differences in mathematics learning outcomes, but readers should keep in mind that using a fixed trait based on a single dimension can lead to a cultural deficit model
100 MATHEMATICS LEARNING IN EARLY CHILDHOOD (Lubienski, 2007). Racial/ethnic groups are heterogeneous, and children in particular racial/ethnic groups have mathematical knowledge and skills that range from low to high mastery levels. Generally, African American, Hispanic, and American Indian/Alaska Native children achieve at lower levels than their white peers in matheÂ matics (National Center for Education Statistics, 2007). Few data exist on early childhood mathematics teaching and learning in relation to race/Â ethnicity, but one can extrapolate from K-12 studies. Findings suggest that this achievement disparity is related to differences in mathematics learning before school entry and fewer meaningful pedagogical experiences once children of color enter school (Magnuson and Waldfogel, 2008). For ex- ample, the National Assessment of Educational Progress (NAEP) survey data show that fourth grade black and Hispanic students and those with low SES report that mathematics mainly consists of memorizing facts, a belief that is negatively correlated with achievement even after controlling for race/ethnicity and SES (Lubienski, 2006, 2007). Furthermore, teachersâ reports indicate that black and Hispanic children were more likely to be routinely assessed with multiple-choice tests than white students (Lubienski, 2006). These practices do not represent the best pedagogy for high-Âquality mathematics education (National Council of Teachers of Mathematics, 2000). Teachers who build on childrenâs everyday mathematical experiences promote genuine mathematics learning (Civil, 1998; Ladson-Billings, 1995). For example, Ladson-Billings (1995) found that urban and suburban stu- dentsâ community experiences shaped the way they approached a math- ematics problem-solving task and that studentsâ differing approaches to learning could be used by teachers to inform their instruction. Instructional practices that extend childrenâs out-of-school experiences are more likely to produce meaningful mathematics learning. English Language Learners Surprisingly little research has examined the mathematics performance of English language learners. Findings for other subject areas show that children who have limited proficiency in English perform more poorly than their native English-speaking peers in other academic subjects (McKeon, 2005). A major issue for educating English language learners (ELL) is the language of instruction (Barnett et al., 2007; Genesee et al., 2006). In re- search conducted by Barnett and colleagues (2007) with 3- and 4-year-olds, they tested whether children in a two-way immersion (English and Spanish) or those in English-only programs made gains in English language measures of mathematics, vocabulary development, and literacy. They found that children in both types of programs made gains on all academic measures
DEVELOPMENTAL VARIATION, SOCIOCULTURAL INFLUENCES 101 and the two-way immersion classrooms saw improvements in Spanish lan- guage development for both ELL and English-speaking children without losses to English language learning (Barnett et al., 2007). It is important to note that classrooms in both types of program employed a licensed teacher and an assistant with a child development associate credential. A review of the K-12 literature on the language of instruction provides evidence that conflicts with the findings of Barnett and colleagues; specifically, Lindholm- Leary and Borasato (2006) suggest that bilingual education may be related to more positive educational outcomes for older ELL students. Given these disparate findings, additional research in high-quality early childhood set- tings on this topic is warranted. One of the few studies focused specifically on mathematics competence with this population of students suggests there may not be performance dif- ferences in mathematics. Secada (1991) found that first grade Hispanic stu- dents were not at a disadvantage to their native English-speaking peers in solving addition and subtraction word problems. However, with the grow- ing number of ELL in the student population, it vital that more attention be paid to the relationship between language status and early mathematics learning so that early childhood education can effectively accommodate and support these children. INFLUENCE OF CONTEXT AND EXPERIENCE As noted in the previous section, research has identified consistent, average differences in mathematics competence and performance depend- ing on membership in a particular social group. Why group membership is linked to such differences is a complicated question. Research suggests that early experiences play an important role in shaping the observed dif- ferences. In this section we explore the contributions of context and early experience. We begin with a general discussion of the role of families in shaping early experience, including parentsâ knowledge and beliefs about mathematics, and the support they provide for mathematics through en- gagement in mathematics activities. We then look more specifically at how differences in experiences at home are linked to the observed SES differ- ences in performance. Finally, we consider the role of language in math- ematics learning. Role of Families Families are one of the critical social settings in which children develop and learn (Bronfenbrenner, 2000; Iruka and Barbarin, 2008). Families influ- ence childrenâs development in many ways, including parenting practices, provision of resources, interactions with school, and involvement in the
102 MATHEMATICS LEARNING IN EARLY CHILDHOOD community (Weiss, Caspe, and Lopez, 2006; Woods and Kurtz-Costes, 2007). Parents have different attitudes, values, and beliefs in raising young children, which result in difference emphasis on educational activities in the home. Families support mathematics learning through their activities at home, conversations, attitudes, materials they provide to their children, expectations they have about their performance, the behaviors they model, and the games they play. Parents also build connections with their childrenâs educational settingsâall of which can shape childrenâs early mathematics development. Parentsâ Knowledge and Beliefs About Early Childhood Mathematics Although there are only a few empirical studies about parental beliefs and behaviors related to early mathematics, those that exist suggest that parents place more importance on literacy development (Barbarin et al., 2008). Barbarin and colleagues examined the beliefs of parents whose children were enrolled in public prekindergarten regarding the skills chil- dren need to be prepared for school. Mathematical skills and such tasks as counting were rated less important than other social and cognitive tasks. Specifically, language/early literacy was mentioned 50 percent of the time, whereas numeracy was mentioned only 3.5 percent of the time (Barbarin et al., 2008). Similarly, Cannon and Ginsburg (2008) found that mothers thought it was more important that their children learn daily living skills and develop language skills in preschool than that their children learn mathematical skills. Most mothers in the study reported they themselves spent more time teaching their children language skills than mathematics skills at home. Engagement in Mathematics Activities Childrenâs mathematical competence is supported and shaped by the math-related activities they engage in as part of their daily lives (Benigno and Ellis, 2008). Parenting practices in which parents engage children in conversations about number concepts, play with puzzles and shapes, en- courage counting, and use number symbols to represent quantity in their interactions in the physical world can facilitate mathematics learning (see Box 4-1 for examples of how parents can engage children in mathematics activities). Acquiring mathematics knowledge involves more than learning numbers. It also includes learning shapes and patterns. It is facilitated by conversations about what children are doing when they compute, solve puzzles, and develop patterns and discussions of why they took a particular approach to a problem. In fact, one study demonstrates how parents and their children can engage in mathematics-related activities. In a groundbreaking study of
DEVELOPMENTAL VARIATION, SOCIOCULTURAL INFLUENCES 103 BOX 4-1 Supporting Childrenâs Mathematics at Home Parents play an important role in supporting mathematics learning through the mathematics-related activities in which they engage their children. Incorporating mathematics-focused activities during play is one strategy for enhancing math- ematics. Another is to capitalize on situations in which mathematics is a natural part of everyday tasks, such as grocery shopping or cooking. During daily activi- ties, parents can: â¢â Observe their children carefully, seeing what they do and encouraging and extending their fledgling use of number symbols and processing. â¢â Say the number word list. For example, they can count small food items or the number of cups at the table. â¢â Ask children to tell them about their problem solving. For example, they can ask âWhat did you mean by that?â or âWhy did you do it that way?â â¢â Engage in activities that involve playing with blocks, building things, and board games. Given the prevalence of the Internet, television, and videogames in the lives of children, even young children (for a review, see Fisch, 2008), these means of communication provide interesting opportunities for impacting early Âmathematics skills. Fisch (2008) provides a review of existing media that include a math- ematical component. These include television shows, such as Sesame Street; m Â athematics-based software games, such as Building Blocks and Millieâs Math House; websites that include mathematics content, such as that of Sesame Street and Disney; and electronic, interactive toys. The Internet can be a tool to help families devise mathematics-related activi- ties for their young children. Such websites as FAMILY MATH, from the Lawrence Hall of Science at the University of California, Berkeley, can provide this kind of help. Although there are no effectiveness data available for this website, FAMILY MATH offers fun activities that maintain mathematical integrity and uses inex- pensive materials that families may already have at home (see http://sv.berkeley. edu/showcase/pages/fm_act.html). early childhood mathematics in family contexts, Saxe and colleagues (1987) found that many of the children in the 78 families they studied, both low and middle income, were spontaneously engaging in number-related activi- ties (counting toys, using numbers in play, etc.), but the nature of their nu- merical knowledge and environment differed. Mothers in the study reported that both they and their children had a high level of interest in number play, but middle-income children performed better than low-income children on both the cardinality and arithmetic tasks. There are numerous opportunities on a daily basis for children and families to explore mathematical terms and concepts. These include meal- times, shopping, playtime, sports, television, and reading (Benigno and Ellis, 2008). In fact, Blevins-Knabe and Musun-Miller (1996) provide evidence
104 MATHEMATICS LEARNING IN EARLY CHILDHOOD to support the effects of parental modeling, reporting a relation between parental participation in number activities and childrenâs involvement in similar activities. Moreover, they found that parental reports of childrenâs number activities at home predicted their scores on a standardized test of early mathematical ability. Several studies suggest that exposure to the language and symbol sys- tem of mathematics powerfully extends the universal starting points of childrenâs quantitative knowledge and contributes to observed differences in mathematics competence. This is true in terms of exposure to the language of mathematics in preschool (Klibanoff et al., 2006) as well as at home be- tween ages 14 and 30 months (Levine et al., in preparation). These studies show that the range of number words used in these settings is enormous. For example, in the home study, a longitudinal project in which families were visited every 4 months for five 90-minute sessions during which they were asked to go about their normal activities, the use of number words ranged from a low of 3 to a high of 175 instances. Similarly, in the class- room studies, the amount of number input provided by teachers during a 1-hour period that included circle time ranged from 1 to 104 coded instances. While research suggests that families do incorporate mathematics into their everyday lives, they may also need reminders of the importance of mathematics. An observational study of 39 preschoolers and their fami- lies (Tudge and Doucet, 2004) found that the children engaged in a very low rate of explicit mathematics lessons over the course of a day and also demonstrated low levels of mathematics-related play. Of the mathematics lessons that were observed, the most common were lessons involving num- bering, and the most common types of mathematical play involved toys that featured numbers (puzzles, computer programs, etc.). Furthermore, parents may overestimate their childrenâs mathematical skills. Fluck and colleagues (2005) found that parents believed their children had a much better grasp of the concept of cardinality (beyond mere counting) than the children actually displayed. Differences in Childrenâs Experiences and Learning Opportunities as a Function of Socioeconomic Status Evidence suggests that SES differences in childrenâs mathematics com- petence are linked to parallel differences in experiences provided in the home. For parents in some low-SES families, involvement in fostering the acquisition of mathematics skills in their children may be hampered by multiple factors. Poverty and uncertainty related to inadequate resources and residential instability can easily become all-consuming, leaving room for little else. Parents in low-SES families, though concerned about their
DEVELOPMENTAL VARIATION, SOCIOCULTURAL INFLUENCES 105 childrenâs education, may feel less ready to assist them due to limitations in their own education, the strains of inadequate financial resources, unmet mental health needs, and specific discomfort with their own mathematical skills and a lack of awareness of the importance of early mathematics devel- opment (for research on the effects of poverty on parenting see, e.g., Knitzer and Lefkowitz, 2006; McLoyd, 1990; see Clements and Sarama, 2007, for a specific discussion of low-income families and mathematics). Research shows that low-income parents provide fewer mathematics activities than middle-class parents (Starkey et al., 1999). This includes free activities, such as those that are integrated into everyday experiences and made-up games, suggesting that, to some extent, lack of financial resources does not explain the difference. Starkey and Klein (2008) suggest that the difference may instead stem from educational background and exposure to mathematics courses. The difference may be resource-based as well. Ramani and Siegler (2008), in a study of board game activities, found that, although 80 percent of middle-class preschool-age children reported play- ing one or more board games outside preschool, only 47 percent of Head Start children did so. However, such board games could easily be made and used at home. It is also vital to remember that, in many cases, children and families from low-SES backgrounds are involved with many more agencies and programs than their more well-off peers. âExploring the contribution of these additional settings is important because interpreting SES effects as emanating exclusively from the family or the child means that policy and program interventions may focus too narrowly as they attempt to improve the educational outcomes of low-SES childrenâ (Aikens and Barbarin, 2008, p. 236). Policy makers, researchers, and practitioners should not neglect the importance of the interactions and experiences of the multiple contexts and the nature of development in everyday life. Thus, at the level of a mother and child interacting in a larger social context unique to cul- tural environments, the entire dynamic may influence a childâs learning and specifically reinforce or hinder the development of mathematical thinking and understanding. The SES gap prior to preschool entry suggests that the home environ- ment plays a major role, yet it is important to note that formal preschool programs do not appear to be ameliorating it. In fact, the gap widens during the preschool years. âIn the United States, neither the home nor preschool learning environments of low-SES children provide sufficient enrichment to close or even maintain early SES-related differences in mathematical knowledgeâ (Starkey and Klein, 2008, p. 266). The issue of how to better support low-income children in mathematics and address the gap is taken up in detail in Chapter 7.
106 MATHEMATICS LEARNING IN EARLY CHILDHOOD Role of Language Languages vary in the ways they represent mathematical concepts. This variation appears to be linked to variation in childrenâs mathematics learning. For example, several recent studies have shown that characteris- tics of speakersâ language influence the quantitative skills of children and adults. One set of studies provides evidence that variations in the structure of a morphological marker, which refers to a language element that iden- tifies quantity in different languages, is associated with the age at which children learn the meaning of specific cardinal numbers. That is, children who speak a language that marks the singular-plural distinction through a morphological marker (e.g., the s on the end of dogs, which indicates that the word is plural, is the morphological marker) acquire the meanings of small cardinal numbers sooner than children whose language does not make such a distinction (e.g., LeCorre, Li, and Lee, 2004; Li et al., 2003; Sarnecka et al., 2007). Even more strikingly, recent evidence has shown that adults in cultural groups with few number words perform worse than adults from cultural groups with more elaborated number systems in matching set sizes, performing arithmetic operations, and other cognitive tasks requiring knowledge of exact numbers (Gordon, 2004; Pica et al., 2004). There is also a large body of evidence regarding the implications of number naming systems for mathematics learning. Language Differences in Number Names Language differences in number names have received in-depth attention in the literature. Such differences appear to be linked to the ease with which children learn to count, an essential task during early childhood. Names and symbols for numbers can be (and have been) generated according to a bewildering variety of systems (see Ifrah, 1985; Menninger, 1958/1969). Because the base-ten system is so familiar and widespread and because humans have 10 fingers, it may appear that the development of a base-ten system is somehow natural and inevitable. Historically, base 4 and base 8 systems were also common (Menninger, 1958/1969). However, most mod- ern languages now use systems that are organized around a base of 10, although languages vary in the consistency and transparency of that struc- ture. For example, number words in English, Spanish, and Chinese differ in important ways. In all three languages, number names can be described to a first approximation as a base-ten system, but the languages differ in the clarity and consistency with which the base-ten structure is reflected in actual number names. Representations for numbers from 1 to 9 consist of an unsystemati- cally organized list. There is no way to predict that â5â or âfiveâ or âwuâ comes after â4,â âfour,â and âsi,â in the Arabic numeral, English, or
DEVELOPMENTAL VARIATION, SOCIOCULTURAL INFLUENCES 107 Chinese systems, respectively. Names for numbers above 10 also diverge in interesting ways among the three languages. The Chinese number-naming system maps directly onto the Hindu-Arabic number system used to write n Â umerals. For example, a word-for-word translation of âshi qiâ (17) into English produces âten-seven.â English has unpredictable names for â11â and â12â that bear only a historical relation to âoneâ and âtwoâ from the Old Saxon ellevan (one left over) and twelif (two left over) (Menninger, 1958/1969). Whether the boundary between 10 and 11 is marked in some way is very significant, because this is the first potential clue to the fact that number names are organized according to a base-ten system. English names for teen numbers beyond twelve do have an internal structure, but this relation is obscured by phonetic modifications of many of the elements from those used for 1 through 10 (e.g., âtenâ becomes âteen,â âthreeâ becomes âthir,â and âfiveâ becomes âfifâ). Furthermore, the order of formation reverses place value compared with the Hindu-Arabic and Chinese systems (and with English names above 20), naming the smaller value before the larger value (e.g., say âfourteenâ but write 14 with the 4 second). Spanish follows the same basic pattern for English to begin the teens, although there may be a clearer parallel between âuno, dos, tresâ and âonce, doce, treceâ than between âone, two, threeâ and âeleven, twelve, thirteen.â The biggest difference between Spanish and English is that, after 15, number names in Spanish abruptly take on a different structure. Thus, the name for 16 in Spanish âdiez y seisâ (literally âten and sixâ), follows the same basic structure as do Arabic numerals and Chinese number names (starting with the tens value and then naming the ones place), rather than the structure used by teens names in English from 13 to 19 and by teens names in Spanish from 11 to 15 (starting with the ones place and then naming the tens value). Above 20, all these number-naming systems converge on the Chinese structure of naming the larger value before the smaller one, consistent with the order of writing the values in numerals. Despite this convergence, the systems continue to differ in the clarity of the connection between decade names and the corresponding unit values. Chinese numbers are consistent in forming decade names by combining a unit value and the base (10). Decade names in English and Spanish generally can be derived from the name for the corresponding unit value, with varying degrees of phonetic modification (e.g., âfiveâ becomes âfifâ in English as in fifty rather that fivety, âcincoâ becomes âcincuentaâ in Spanish) and some notable exceptions, primarily the special name for twenty (âveinteâ) used in Spanish. Consequences for Learning to Count Although languages differ in the length and complexity of the irregular portion of the system of names that must be learned, in general children
108 MATHEMATICS LEARNING IN EARLY CHILDHOOD must learn quite a few number names prior to coming across data support- ing the induction that they are dealing with an ordered base-ten system of names. Looking at the extent to which differences in learning reflect dif- ferences in counting terms can assess effects of number-naming systems on childrenâs early mathematics. Research on childrenâs acquisition of number names (Fuson, ÂRichards, and Briars, 1982; Miller and Stigler, 1987; Siegler and Robinson, 1982) suggests that children in America learn to recite the list of number names through at least the teens in essentially a rote learning task. When first counting above twenty, U.S. preschoolers often produce idiosyncratic num- ber names, indicating that they fail to understand the base-ten structure underlying larger number names, often counting âtwenty-eight, twenty- nine, twenty-ten, twenty-eleven, twenty-twelve.â This kind of mistake is extremely rare for Chinese children, indicating that the base-ten structure of number names is more accessible for learners of Chinese than it is for children learning to count in English. The cognitive consequences of the relative complexity of English num- ber names are not limited to obstacles placed in the way of early counting. Speakers of English and other European languages (Fuson, Fraivillig, and Burghardt, 1992; SÃ©ron et al., 1992) face a complex task in learning to write Arabic numerals, one more difficult than that faced by speakers of Chinese (compare the mapping between name and numeral for âtwenty- fourâ with that for âfourteenâ in the two languages). Work by Miura and her colleagues (Miura, 1987; Miura and Okamoto, 1989; Miura et al., 1988, 1993) suggests that the lack of transparency of base-ten markings in English has conceptual consequences as well. They have found that speak- ers of languages whose number names are patterned after Chinese (includ- ing Korean and Japanese) are better able than speakers of English and other European languages to represent numbers using base-ten blocks and to perform other place-value tasks. Because school arithmetic algorithms are largely structured around place value, this indication that the complexity of number names affects the ease with which children acquire this basic concept is a finding with real educational significance. When learning to count, children must acquire a combination of con- ventional knowledge about number names (they must learn their own cultural number word list in order), a conceptual understanding of the mathematics principles that underlie counting, and an ability to apply this knowledge to mathematical problem solving. Language differences during preschool appear to be limited to the first aspect of learning to count. For example, Miller and colleagues (1995) found no differences between Chi- nese and U.S. preschoolers in the extent to which they violated counting principles when counting objects, or in their ability to use counting to pro- duce sets of a given size in the course of a game. The effects of differences
DEVELOPMENTAL VARIATION, SOCIOCULTURAL INFLUENCES 109 in number-name structure on early mathematical development appear to be very specific to those aspects of mathematics that require one to learn and use these symbol systems. These effects have implications for learning Arabic numerals and thus for acquiring the primary symbol system used in school-based mathematics. The nature and timing of differences in early counting between Chinese- speaking and English-speaking preschoolers correspond to predictions based on the morphology of number names. Evidence from object counting indicates that these differences are also limited to aspects of counting that involve number naming. Miller and colleagues (1995) looked at childrenâs object counting for sets that were small (3-6 items), medium (7-10 items), and large (14-17 items). They found that Chinese-speaking children were significantly more likely to report the correct number word for a set than English-speaking children, but this was entirely due to the greater likelihood of Chinese children to correctly recite the sequence of names. The task of completely coordinating saying number words and designating objects in counting is quite difficult for many young preschoolers, and equally so for U.S. and Chinese children: 37 percent of U.S. and 38 percent of Chinese pre- schoolers either pointed to an object and did not produce the number name or the reverse. Double counting or skipping objects was even more com- mon, but again did not differ between the Chinese and U.S. preschoolers. Consequences for Using the Base-ten Structure in Problem Solving The structure of number names is associated with a specific, limited difference in the course of counting acquisition between English-speaking and Chinese-speaking children. One area in which there may be conceptual consequences of these linguistic differences is in childrenâs understanding of the base-ten principle that underlies the structure of Arabic numerals. This structure is a feature of a particular representational system rather than a fundamental mathematical fact, but it is a feature that is incorporated into many of the algorithms children learn for performing arithmetic and thus is a powerful concept in early mathematical development. Because English number names do not show a base-ten structure as consistently or as early in the count sequence as do Chinese number names, English-speaking chil- drenâs conceptual understanding of this base-ten structure may be delayed compared with their Chinese-speaking peers. Miura and her colleagues (Miura, 1987; Miura and Okamoto, 2003; Miura et al., 1993) have looked at the base-ten understanding of two groups of first grade children: speakers of East Asian languages, whose number-naming systems incorporate a clear base-ten structure, usually based on Chinese, and speakers of European languages, which generally do not show a clear base-ten structure in their number names. The primary
110 MATHEMATICS LEARNING IN EARLY CHILDHOOD task used is asking children to represent the cardinal value associated with a given number name using sets of blocks representing units and tens. Chil- dren whose native language is Chinese, Korean, or Japanese are consistently more likely to represent numbers as sets of tens and ones as either a first or second choice than are children whose native language is English, French, or Swedish. Ho and Fuson (1998) compared the performance of Chinese-speaking preschool children in Hong Kong with English-speaking children in Britain and the United States. They found that half of the Chinese-speaking 5-year- olds (but none of the English-speaking children) who could count to at least 50 were able to take advantage of the base-ten structure of number names to quickly determine the answer to addition problems of the form â10 + n = ?,â compared with other problems. Fuson and Kwon (1992) argued that the Chinese number-naming structure facilitates the use of a tens-complement strategy for early addition. In this approach, when adding numbers whose sum is greater than 10 (e.g., 8 + 7), the smaller addend is partitioned into the tens-complement of the first addend (2) and the remainder (5); the answer is 10 plus that remainder (10 + 5). In Chinese-structured number- naming systems, the answer corresponds to the result of the calculation (âshi wuâ â â10 5â); in English, there is an additional step as the answer is converted into a different number name (âfifteenâ). Fuson and Kwon reported that most Korean first graders they tested used this method before it was explicitly taught in school. Explicit instruction may be required for English-speaking children, but there is evidence that it can be quite success- ful, even with children from at-risk populations. Fuson and her colleagues (Fuson, Smith, and Lo Cicero, 1997) report success with explicitly teaching low-SES urban first graders about the base-ten structure of numbers, with the result that their end-of-year arithmetic performance approximated that reported for East Asian children. LEARNING DISABILITIES IN MATHEMATICS Mathematics learning disabilities appear in 6 to 10 percent of the el- ementary school population (Barberisi et al., 2005). Many more children struggle in one or more mathematics content area at some point during their school careers (Geary, 2004). Although less research has been devoted to mathematical than to reading disabilities (Geary and Hoard, 2001; Ginsburg, 1997), considerable progress has been made over the past two decades with respect to understanding the nature of the mathematics dif- ficulties and disabilities that children experience in school (Gersten, Jordan, and Flojo, 2005).
DEVELOPMENTAL VARIATION, SOCIOCULTURAL INFLUENCES 111 Characteristics of Learning Difficulties Poor computational fluency is a signature characteristic of mathematics learning disabilities in elementary school (e.g., Geary, 2004; Hasselbring, Goin, and Bransford, 1988; Jordan and Montani, 1997; Jordan, Hanich, and Kaplan, 2003a, 2003b; Ostad, 1998; Russell and Ginsburg, 1984). Computational fluency refers to accurate, efficient, and flexible computa- tion with basic operations. Weak knowledge of facts reduces cognitive and attentional resources that are necessary for learning advanced mathematics (Goldman and Pellegrino, 1987). Computational fluency deficits can be reliably identified in the first few years of school and, if not addressed, are very persistent throughout elementary and middle school (Jordan, Hanich, and Kaplan, 2003b). Children around the world move through a learning path of levels of solution methods for addition and subtraction problems. These levels be- come progressively more abstract, abbreviated, embedded, and complex. As they move through the levels, many children use a mix of strategies that vary according to number size and aspects of the problem situation (Geary and Burlinghman-Dubree, 1989; Siegler and Jenkins, 1989; Siegler and Robinson, 1982; Siegler and Shipley, 1995). In contrast, young children with a mathematics learning disability rely on the most primitive Level 1 methods for extended periods in elementary school, do not use efficient counting procedures (e.g., counting on from the larger addend), and make frequent counting errors while learning to add and subtract (Geary, 1990). They also lag behind other children in the accuracy and linearity of their number line estimates (Geary et al., 2007). Researchers have differentiated children with a specific mathematics learn- ing disability from those with a comorbid learning disability in both math- ematics and reading. Jordan and colleagues (Hanich et al., 2001; Jordan, Hanich, and Kaplan, 2003a; Jordan, Kaplan, and Hanich, 2002) as well as other researchers (e.g., Geary, Hamson, and Hoard, 2000; Landerl, Bevan, and Butterworth, 2004) suggest that the nature of the mathematical deficits is similar for both groups, although children with the comorbid condi- tion show lower performance overall. What differentiates children with a mathematics-only disability from those with combined mathematics and reading learning disabilities is that the former group performs better on word problems in mathematics, which depend on language comprehension as well as calculation facility. The potential for catching up in mathematics is much better for children with a mathematics-only disability, who can exploit their relative strength in general language to compensate for their deficiencies with numbers. Some research shows that mathematics learning disabilities can be traced to early weaknesses in number, number relationships, and number
112 MATHEMATICS LEARNING IN EARLY CHILDHOOD operations as opposed to more general cognitive deficits (e.g., Gersten et al., 2005; Malofeeva et al., 2004). Weak number competency is reflected in poorly developed counting procedures, slow fact retrieval, and inaccurate computation, all characteristics of the disability (Geary et al., 2000; Jordan, Hanich, and Kaplan, 2003a). Skill with number combinations is tied to fundamental number knowledge (Baroody and Rosu, 2006; Locuniak and Jordan, in press). Accurate and efficient counting procedures can lead to strong connections between a problem and its solution (Siegler and Shrager, 1984). Developmental dyscalculia, a severe form of mathematics disability that has a known neurological basis, is explained more by domain-specific impairments in number knowledge than by domain-general deficits related to memory, spatial processing, or language (Butterworth and Reigosa, 2007). Although debate continues about the underpinnings of mathemat- ics learning disabilities and diagnostic criteria (e.g., Geary et al., 2007), weakness in number sense appears to be a common theme in the literature. This finding has instructional implications for young childrenâs mathematics education. Specifically, early interventions that focus on number sense have the potential to improve childrenâs mathematics learning outcomes. Helping High-Risk Children Early number competencies serve as a foundation for learning formal mathematics (Griffin et al., 1994; Miller, 1992). Deficits in these can pre- vent children from benefiting from formal mathematics instruction when they enter school, regardless of whether they are associated with environ- mental disadvantages or with genuine learning differences or disabilities (Baroody and Rosu, 2006; Griffin, 2007). In a recent study, Jordan and colleagues (in press) found that poor mathematics achievement is mediated by low number sense regardless of childrenâs social class. That is, deficits in number sense are a better predictor of poor mathematics achievement than SES when all else is equal. Implications of this work suggest that chil- dren from low-income backgrounds and those with mathematics difficul- ties would benefit from a mathematics intervention during the early years (Jordan et al., in press). Number competencies appear to have neurological origins, with their core components (e.g., subitization and approximate number representa- tions) developing without much formal instruction (Berch, 2005; Dehaene, 1997; Feigenson, Dehaene, and Spelke, 2004). These early foundations provide support for learning more complex number skills involving number words, number comparisons, and counting. Children with mathematics dif- ficulties seem to have problems with the symbolic system of number, rather than the universal analog magnitude system. Knowledge of the symbolic number system is heavily influenced by experience and instruction (Geary,
DEVELOPMENTAL VARIATION, SOCIOCULTURAL INFLUENCES 113 1995; Levine et al., 1992). Engaging young children in number activities (e.g., a mother or preschool teacher asking a child to give her 4 cookies) and simple games (e.g., board games that emphasize 1-to-1 correspondences, counting, and moving along number paths) are important for strengthen- ing foundations and building conventional number knowledge (Gersten et al., 2005, Klibanoff et al., 2006; Levine et al., in preparation). Case and Griffin (1990) report that number sense learning is closely associated with childrenâs home experiences with number concepts (e.g., reading number books with children). Moreover, efforts to teach number-related skills to high-risk kindergartners show promise for improving mathematics achieve- ment (Griffin et al., 1994). In a recent study, Ramani and Siegler (2008) showed that playing a number board game that involved counting on squares on a number path improved the performance of 5-year-olds from low-income backgrounds on counting, numeral identification, numerical magnitude estimation, and number line estimation, and that the gains held after a follow-up several weeks later. Importantly, children playing this game said the number words written on the squares as they counted on one or two more, rather than saying âoneâ or âtwoâ as they counted on. Play- ing games to help children master basic number, counting, and arithmetic concepts and skills has long been advocated by mathematics educators (e.g., Baroody, 1987; Ernest, 1986; Wynroth, 1986)âa proposition that is sup- ported by research (for reviews, see, e.g., Baroody, 1999; Bright, Harvey, and Wheeler, 1985). The effects of weaknesses in early mathematics, if not addressed, are likely to be felt throughout the school years and beyond. There is good reason to believe that early intensive instruction, both at home and at school, will give children the background they need to achieve at grade level in elementary school mathematics and help âshape the course of their mathematical journeyâ (Griffin, 2007, p. 392). REFERENCES AND BIBLIOGRAPHY Aikens, N.L., and Barbarin, O. (2008). Socioeconomic differences in reading trajectories: The contribution of family, neighborhood, and school contexts. Journal of Educational Psychology, 100(2), 235-251. Arnold, D.H., Fisher, P.H., Doctoroff, G.L., and Dobbs, J. (2002). Accelerating math develop- ment in Head Start classrooms. Journal of Educational Psychology, 94(4), 762-770. Baenninger, M., and Newcombe, N. (1989). The role of experience in spatial test performance: A meta-analysis. Sex Roles, 20, 327-344. Barbaresi, M.J., Katusic, S.K., Colligan, R.C., Weaver, A.L., and Jacobsen, S.J. (2005). Math learning disorder: Incidence in a population-based birth cohort, 1976-1982, Rochester, Minn. Ambulatory Pediatrics, 5(5), 281-289. Barbarin, O.A., Early, D., Clifford, R., Bryant, D., Frome, P., Burchinal, M., Howes, C., and Pianta, R. (2008). Parental conceptions of school readiness: Relation to ethnicity, socioeconomic status, and childrenâs skills. Early Education & Development, 19(5), 671-701.
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